Gaussianrelated

Gaussianrelated denotes topics, results, and objects that are tied to Gaussian distributions or Gaussian processes in probability, statistics, and related fields. In probability theory, Gaussianrelated covers the ordinary Gaussian (normal) distribution N(mu, sigma^2), its properties, and common transformations.

Key features include symmetry, defined mean mu and variance sigma^2, the standard normal Z ~ N(0,1), and moment generating and characteristic functions. The moment generating function is M(t) = exp(mu t + 1/2 sigma^2 t^2) and the characteristic function is φ(t) = exp(i mu t - 1/2 sigma^2 t^2). Gaussian sums remain Gaussian, and linear transformations preserve Gaussianity.

Multivariate Gaussian distributions extend these ideas to vectors X ~ N(mu, Σ) with density proportional to exp(-1/2 (x-μ)^T Σ^{-1} (x-μ)). A Gaussian process is a collection of random variables where every finite subset is multivariate Gaussian, fully specified by a mean function m(t) and a covariance function k(s,t).

Gaussian integrals are fundamental tools: ∫ exp(-x^2) dx = sqrt(pi) in one dimension, and the general form ∫ exp(-1/2 x^T Σ^{-1} x) dx = (2π)^{n/2} det(Σ)^{1/2} in n dimensions.

Applications include statistics, data analysis, signal processing, machine learning (notably Gaussian processes for regression and classification), and kernel methods using Gaussian kernels. In image and signal processing, Gaussianrelated terms appear in noise models, smoothing (Gaussian blur), and density estimation.

The term Gaussianrelated is descriptive rather than a formal designation, used to indicate material connected to Gaussian ideas.